A physical-constraints-preserving arbitrary Lagrangian–Eulerian discontinuous Galerkin scheme on adaptive unstructured meshes for two-component flows

In this paper, a physical-constraints-preserving arbitrary Lagrangian–Eulerian (ALE) discontinuous Galerkin (DG) scheme on adaptive unstructured meshes for two-component flows is developed. In our scheme, a conservative equation related to the volume fraction model is coupled with Euler equations for describing the dynamics of fluid. The mesh velocity in the ALE framework is given by the adaptive mesh method, which can automatically concentrate the nodes near the regions with the large gradients of some variables. With the help of this adaptive mesh, the numerical dissipation near material interfaces caused by the volume fraction model can be greatly reduced; meanwhile, the resolution of the solution near the wave structures can be improved effectively. For ensuring the computational robustness and physical properties of fluid under low density and so on, our scheme will use the appropriate constraints and a physical-constraints-preserving limiter to keep the positivity of density and specific internal energy and the boundness of volume fraction. This concise scheme can be applied to the simulations of two-component flows efficiently with the physical-constraints-preserving (bound-preserving and positivity-preserving) property. Some examples are used to demonstrate the accuracy, essentially non-oscillatory property, and physical-constraints-preserving property of our physical-constraints-preserving ALE-DG scheme.